Convergence Analysis of an Extended Krylov Subspace Method for the Approximation of Operator Functions in Exponential Integrators

We analyze the convergence of an extended Krylov subspace method for the approximation of operator functions that appear in exponential integrators. For operators, the size of the polynomial part of the extended Krylov subspace is restricted according to the smoothness of the initial data. This restriction for the continuous operator has a significant influence on the approximation of matrix functions evaluated for matrices stemming from space discretizations of the continuous operator. We prove convergence of the method for the continuous operator and, in the discrete case, this leads to a convergence independent of the norm of the discretized operator uniformly over all possible grids. The analysis is illustrated by numerical experiments.